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2 years ago

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There are twelve shirts in your closet, five blue and seven green. Four of the blue shirts and one of the green shirts fit well.

The others are too big. You randomly select a shirt to wear. It is blue or is too big. Find the probability of this occuring

1 answer:

uranmaximum [27]2 years ago

7 0

Answer:

P (blue or too big) = 11 / 12

Explanation:

A two way table permits to represent the situation clearly and find the answser is an easy way.

The two way table is that results from the given information is:

Initial information:

Blue Green Total

Fit well 4 1 ?

Too big ? ? ?

Total 5 7 12

Now by subtraction and addition you can fill the unknown fields:

Blue Green Total

Fit well 4 1 5

Too big 1 6 7

Total 5 7 12

Now, you can see that there are 5 blue and 7 too big shirts, but the total number of them is 5 + 7 - 1 (because one of the too big shirts is also blue).

Then, the number of shirts being blue or too big is 5 + 7 - 1 = 11.

And the probability is :

P(blue or too big) = number of blue or too big / total number of shirts

P (blue or too big) = 11 / 12 ← answer

You can also realize that there is only 1 green shirt that fits well, and that is the only option of not being blue of too big, so 1 /12 is the probability of not being blue or too big, and its complement (being blue or too big) is 1 - 1/12 = 11 / 12.

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You are given that a wheel has a radius of 2 feet and a spin rate of 10 revolutions per minute. Describe how you would determine

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Answer:

About 126 ft/min

Step-by-step explanation:

We know r = 2 feet, and that the wheel has an angular speed of 10 rpms, or 10 revolutions per minute. Our goal is to get the speed in ft/sec.

10 rev 1 minute 2π radians 2 feet

----------- × ---------------- × ------------------ × ------------

1 minute 60 seconds 1 revolution 1 radian

= 10 * 2π * 2 = 126 ft/minute

The answer is reasonable. Another, simple approach to this would be using the formula 2π * r * rpm. If so, 2π * 2 * 10 which would be 126 ft/minute.

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Step-by-step explanation:

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Solve the recurrence relation: hn = 5hn−1 − 6hn−2 − 4hn−3 + 8hn−4 with initial values h0 = 0, h1 = 1, h2 = 1, and h3 = 2 using (

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(a) Suppose $h_n=r^n$ is a solution for this recurrence, with $r\neq0$ . Then

$r^n=5r^{n-1}-6r^{n-2}-4r^{n-3}+8r^{n-4}$
$\implies1=\dfrac5r-\dfrac6{r^2}-\dfrac4{r^3}+\dfrac8{r^4}$
$\implies r^4-5r^3+6r^2+4r-8=0$
$\implies (r-2)^3(r+1)=0\implies r=2,r=-1$

So we expect a general solution of the form

$h_n=c_1(-1)^n+(c_2+c_3n+c_4n^2)2^n$

With $h_0=0,h_1=1,h_2=1,h_3=2$ , we get four equations in four unknowns:

$\begin{cases}c_1+c_2=0\\-c_1+2c_2+2c_3+2c_4=1\\c_1+4c_2+8c_3+16c_4=1\\-c_1+8c_2+24c_3+72c_4=2\end{cases}\implies c_1=-\dfrac8{27},c_2=\dfrac8{27},c_3=\dfrac7{72},c_4=-\dfrac1{24}$

So the particular solution to the recurrence is

$h_n=-\dfrac8{27}(-1)^n+\left(\dfrac8{27}+\dfrac{7n}{72}-\dfrac{n^2}{24}\right)2^n$

(b) Let $G(x)=\displaystyle\sum_{n\ge0}h_nx^n$ be the generating function for $h_n$ . Multiply both sides of the recurrence by $x^n$ and sum over all $n\ge4$ .

$\displaystyle\sum_{n\ge4}h_nx^n=5\sum_{n\ge4}h_{n-1}x^n-6\sum_{n\ge4}h_{n-2}x^n-4\sum_{n\ge4}h_{n-3}x^n+8\sum_{n\ge4}h_{n-4}x^n$
$\displaystyle\sum_{n\ge4}h_nx^n=5x\sum_{n\ge3}h_nx^n-6x^2\sum_{n\ge2}h_nx^n-4x^3\sum_{n\ge1}h_nx^n+8x^4\sum_{n\ge0}h_nx^n$
$G(x)-h_0-h_1x-h_2x^2-h_3x^3=5x(G(x)-h_0-h_1x-h_2x^2)-6x^2(G(x)-h_0-h_1x)-4x^3(G(x)-h_0)+8x^4G(x)$
$G(x)-x-x^2-2x^3=5x(G(x)-x-x^2)-6x^2(G(x)-x)-4x^3G(x)+8x^4G(x)$
$(1-5x+6x^2+4x^3-8x^4)G(x)=x-4x^2+3x^3$
$G(x)=\dfrac{x-4x^2+3x^3}{1-5x+6x^2+4x^3-8x^4}$
$G(x)=\dfrac{17}{108}\dfrac1{1-2x}+\dfrac29\dfrac1{(1-2x)^2}-\dfrac1{12}\dfrac1{(1-2x)^3}-\dfrac8{27}\dfrac1{1+x}$

From here you would write each term as a power series (easy enough, since they're all geometric or derived from a geometric series), combine the series into one, and the solution to the recurrence will be the coefficient of $x^n$ , ideally matching the solution found in part (a).

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The binomial (y − 2) is a factor of y2 − 10y + 16. What is the other factor?

PtichkaEL [24]

Answer:

$\left(y-8\right)$

Step-by-step explanation:

$y^2-10y+16$

$=\left(y^2-2y\right)+\left(-8y+16\right)$

$y^2-2y$

$=yy-2y$

$=y\left(y-2\right)$

$-8y+16$

$=-8y+8\cdot \:2$

$=-8\left(y-2\right)$

$=y\left(y-2\right)-8\left(y-2\right)$

$=\left(y-2\right)\left(y-8\right)$

The other factor is:

$\left(y-8\right)$

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Two car rental companies charge an initial rental fee plus a fee per day. The table shows the total costs of renting a car for x

Fynjy0 [20]

Answer:

a) Which rental company charges less per day?

Payless rental company

b) How many days must you rent a car for the total costs to be the same?

3 days.

Step-by-step explanation:

We have car rental companies

Discount car rentals and Pay less car rentals.

x = number of days

y = Total cost.

a) Let's assume the starting day of renting a car is 2 days

We need to determine the Equation for discount rentals.

For Discount car rentals

2 days = $185

4 days = $265

6 days = $345

8 days = $425

The difference between 2 days = $265 - $185 = $80

If the difference between 2 days = $80

Difference between 1 day = $80/2

= $40

If you rent a car for 1 day,

185 - 40 =$145

Hence, the Equation for payment for Discount rentals =

y = 40x + 105

b) For Pay less car rentals

This equation was given: y = 55x + 60.

y = Total cost, x = number of days

a) 1 day

y = 55(1) + 60

y = $115

b) 2 days

y = 55(2) + 60

y = 110 + 60

y = $170

c) 3 days

y = 55(3) + 60

= 165 + 60

= $225

d)For 4 days

y = 55x + 60.

y = 55 × 4 + 60

y = 220 + 60

y = $280

e) For 5 days

y = 55x + 60.

y = 55 × 5 + 60

y = 275 + 60

y = $335

f) For 6 days

y = 55x + 60.

y = 55 × 6 + 60

y = 330 + 60

y = $380

g) For 7 days

y = 55x + 60.

y = 55 × 7 + 60

y = 385 + 60

y = $445

h) For 8 days

y = 55x + 60.

y = 55 × 8 + 60

y = 440 + 60

y = $500

a) Which rental company charges less per day?

The total coast for renting a car as determined in the above calculation

For 1 day in Discount car rentals = $145

For 1 day in Payless car rentals = $115

From the above calculation, it is Payless rental company

b) How many days must you rent a car for the total costs to be the same?

We equate both equations to each other

y = 40x + 105 = y = 55x + 60.

40x + 105 = 55x + 60

105 - 60 = 55x - 40x

45 = 15x

x = 45/15

x = 3 days.

Therefore, after 3 days, the total cost of renting a car from Discount car rentals or Payless car rentals is the same.

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